In this work, the evolution of entropy production in the Gray-Scott model (P. Gray and S. K. Scott, Chemical Engineering Science 38 (1983) 29-43) is investigated for four representative cases within three regimes. It is revealed that the extremal trends of entropy production for these cases are regime-dependent. In the monostable regime without heterogeneous structures, the steady state follows the minimization of entropy production. For the bistable uniform regime, the steady state with higher entropy production prevails over that with lower entropy production. As to the heterogeneous states that emerge around the bifurcation curves, the entropy production varies in both space and time, showing alternate appearance of states corresponding to minimum and maximum values of entropy production, respectively. The spatiotemporal compromise between maximization and minimization of entropy production appears to govern the corresponding steady states in different regimes. (C) 2016 Elsevier Ltd. All rights reserved.